3-condutivityfermigas

7/28/2019 3-condutivityfermigas

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Chapter 6: Free Electron FermiGas Electrical Conductivity

Chris Wiebe

Phys 4P70


2/15

Free Electron Fermi Gas

Last lecture, we showed that the free electron fermi gas

was successful in describing why Cel = T

This lecture, we will be look at electrical conductivity

Can the free electron model reproduce experimental

resistivities?

The momentum of a free electron is mv = k

What happens when we apply an electromagnetic field

to a system of free electrons?

They feel a Lorentz force:

)

1

( BvcEedt

kd

dt

vd

mF

rrrr

h

rr

+===(CGS units)

Electric charge

Speed of light


3/15

Shift in k-vectors

For now, lets look at the case where B = 0 (well look at

what happens when B 0 later)

So we then have:

When we apply an electric field from some time t = 0 to

some time t, what happens to the k vectors?

dtEe

kdEedt

kdF

r

h

rrr

hr

===

tEe

ktktdEe

kd

ttk

k

r

h

rrr

h

rr

r

== )0()(0

)(

)0(

k-vector shifts in the direction of E


4/15

Fermi Spheres What is happening?

Say the field is applied in the x-direction

Net momentum in kxdirectionFky ky

kx

kx

After the field is applied, the gas feelsa net momentum in the x-direction (all

the states are shifted slightly)

Before a field is applied, there is nonet momentum of the Fermi Gas


5/15

Electrical Conductivity

The electrons all feel a shift in their k-values of:

Now, according to this simple theory, the longer we leave the fieldon, the faster and faster the electrons start to move (the k-values,which are proportional to the momenta, keep on increasing in thex-direction)

Is this observed in the real world?

This would mean that if you apply a field to a copper wire, andcreate an electrical current (movement of electrons), the currentwould grow as a function of time, apparently without a limit

What stops the electrons from moving faster and faster in thiselectric field? (they are accelerating under this force)

h

r

h

rr

tEetFk ==


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Collisions

The reason why we dont see this experimentally is that the

electrons suffer collisions which decrease the velocity

Collisions typically occur with

1. Impurities in the lattice

2. Lattice imperfections (ie. Dislocations, point defects)

3. Phonons

The simplest approximation we can make is that the electronslose all of their kinetic energy after each collision.

In this model, the electrons have a typical collision time , which is

the time that they are accelerated from zero velocity to some

maximum velocity, v, and after another collision, v = 0 again

F


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Back to the Fermi Sphere

Another way of saying this is that in the steady state, the

Fermi sphere is displaced according to the equation we

had before with t =

Fky The incremental (additional)

velocity of the electrons is then:

v = -eE/m

and for a concentration of n electrons,

the current density J (electrons per

unit area per second) is:

J = nqv = ne2E/m

kx


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Ohms law

We have actually just derived Ohms law

The electrical conductivity is defined by Ohms law to be

J = E

On the last slide we showed that

J = ne2E/m

So we can now say that the electrical conductivity is simply: = ne2/m = (ne) (e/m)

And the resistivity is the inverse of the conductivity:

= m/(ne2)

Charge density is ne e/m factor from the acceleration in electric field

Collision time

(note: units are in Ohm cm)Other definition: = E/ J


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Mean Free Path

How do we determine the collision time?

Electrons have a mean free path that they travel before a collision

occurs l = v

At low temperatures, most of the mobile electrons are right at the

Fermi surface, so v = vf(Fermi velocity)

At these temperatures, one can have mean free paths on the orderof ~ 1 cm (!) for very pure crystals (even up to 10 cm for some

extremely pure metals!)

Eg. Copper has l (4K) = 0.3 cm

Compare to the high temperature value: l (300 K) = 3 x 10-6 cm More collisions at high temperatures (as expected). This leads to

shorter collision times, and therefore higher resistivities ( tends to

grow as you increase the temperature)


10/15

Experimental Resistivities

At room temperature (300 K), the electrical resistivity is dominated

by electron collisions with phonons At low temperatures (~ 4 K), it is determined by collisions with

impurities (there arent very many phonons around)

The rates of these collisions are pretty much independent of one

another, so we have:

Matthiessens rule: = L + i

Note: this implies that the collision times are related by:

1/

= 1/

L + 1/

i

Imperfection resistivity

(temp. independent)

Phonon resistivity

(related to the concentrationof phonons, so it is temp. dependent)


11/15

Breakdown of Fermi Electron Gas

Theory At extremely low temperatures for some metals, the resistivity undergoesa remarkable change

Metals such as Zn, Ti, and V superconduct! (at Tc = 0.875 K, 0.39 K, and

5.38 K respectively) This means that the resistivity drops to zero the electron free paths

become infinite!

Kammerlingh Ones was the first one to notice this for Hg at ~ 4.153 K

What is happening here?

Some residualresistivity at T = 0 K


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Superconductivity

In a superconductor, the currents effectively run forever there areno collisions to slow them down (measurements by File and Mills

suggest that the decay time of a supercurrent through a solenoid is

no less than 100 000 years)

Another odd property of superconductors:The Meissner Effect

If a superconducting sample is cooled in a small magnetic field, the

magnetic field lines will be expelled from the sample (due to the

supercurrents forming in a direction to oppose the field, and thereforethe field inside the superconductor is zero)


13/15

Magnetic Levitation

This is what causes the

levitation of magnets above

superconducting samples(the supercurrents form to

counterbalance the

magnetic force, and when

the forces are equal andopposite, the magnet floats)

Potential application:

levitation of magnetic trains

(no friction)

Electrons in magnet, which create afixed magnetic field

Superconducting electrons in sample

(in a direction which counters the magnet to expel the magnetic field)


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Type I and Type II

Superconductors Type I superconductor: A field

can be applied to somemaximum value before it

becomes normal. The fielddoes not penetrate thesuperconductor (Meissnereffect). Most metals belong tothis class (eg. Zn, V, Ti)

Type II superconductor: A fieldcan be applied up to a criticalvalue, HC1, where the fieldlines penetrate the sample.

This is known as the vortexstate. After the field isincreased to HC2, the materialis no longer superconducting.These are the high-Tc

superconductors, likeYBa2Cu3O7.

Penetration of magnetic field lines

in a type II superconductor

Explanation using statistical


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Explanation using statistical

physics What happens to our free electron theory?

The whole reason why this theory works isthat you can only have 2 electrons in eachstate (because electrons have s=1/2 and

they are fermions) But at low enough temperatures, theelectrons are moving slow enough for othereffects to become important (free electrontheory breaks down)

In BCS theory (Bardeen-Cooper-Schrieffer),phonons cause a temporary build up ofpositive charge, which attracts electrons

So, the electrons, which move much fasterthan the phonons, are attracted to oneanother for brief periods of time

They can pair up into integer states

(eg. S = 0 singlet or S = 1 triplet)

If they have integer spin, they are bosons.They can all be in the same energy state

this is often called Bose Condensation. So, they can all be at the same energy they can all move in the same way when anelectric field is applied

e

Total electron spin = 0

(a boson, no longer a fermion)

In BCS theory, phonons can

cause electrons to become attractedto one another

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